On the geometric realization of Albanese's inequality

Abstract

If an algebraic curve C (irreducible and reduced) moving in a family in projective nspace specializes into a curve Co, having associated cycle Z = mlB1 + ... + m,B,, then the geometric genera g, gl,..., gr of C, BI,..., B r respectively and the coefficients m~,..., m, must satisfy a certain inequality (found by Albanese). The realization (or existence) problem asks whether an inequality of this type actually arises from an algebraic family of curves. In this paper some results are obtained concerning the strong version of the problem, where one specifies the cycle Z; with these an affirmative solution of the weak version (where the components B i are not specified) is obtained.